Remark 6.1 In [1]} it is proved that equation (REF ) with \(c_1 \equiv 0\)  is the self-adjoint case. Moreover, A. Mironov (see [2]}) considered the self-adjoint case in the first Weyl algebra, that is \(U\equiv 0\)  in (REF ). He proved the Novikov's conjecture: the existence of \(M\)  in \(\mathcal {C}(L_4)\)  such that \(h(L_4,M)=0\)  for \(h(\lambda ,\mu )=\mu ^2+R_{2g+1}(\lambda )\)  the defining polynomial of a genus \(g\)  curve \(\Gamma \) ; furthermore this operator \(L_4\)  has an order 2 factor at each point of \(\Gamma \) .

